In controlling both Westeros and Western Essos, they would already essentially be a client state.
I think assessing, on his end, just how far we intend to go in Essos is conditional to his future goals. If we say "just stopping at the Three Daughters to fund and build an army to retake my birthright", then we become an important local actor who is actually relevant to keep abreast of and maintain a military and economic alliance with.
If we intend to conquering most of our neighbors, then that basically changes the tone of the conversation by many degrees. And, facing facts, this is what Braavos is facing:
He can't get our tacit and full support (both as a deterrent to enemies and also in practical terms) against Pentos and any aggressors if he does not, in all likelihood, cede some literal and figurative ground. Then, just on that issue, it becomes a more long term concern, because we would share land borders with them.
But that's not where we let it end, no, then we absorb Volantis, Norvos, the Rhoyne included, Qohor. Start a reclamation project for Essaria, fund expeditions for former Sarnori expats or remnant city states to reclaim their heartland (so we can assimilate them), assert our authority over North Valyria, absorb former and New Ghiscar.
There's good news: He does still have leverage. He has trade contacts with the Far East through his various subject Houses, and we favor Braavosi trained administrators. Seeing the way the wind is blowing, he could focus on his strengths to get favorable terms as a client state or protectorate, with essentially free reign within his remit.
Or he could push away someone who already loves his City too much to do it lasting harm, militarily or economically... which would be absurd--he knows what he's been doing thus far. He knows where he gets his bread buttered. He's having us arrange for his political rivals to be assassinated... you don't outsource that to another state unless you expect it to become a superpower in the near future.
Yes, I'm sort of leaning on real-life a bit with that assumption, and the fact this takes place on a death world with extraplanar powers and eldritch abominations trying to muscle into the action sort of muddies the waters, but it's still pretty accurate.
